A Geometric Perspective on the Closed Convex Hull of Some Spectral Sets

Renbo Zhao

doi:10.48550/arxiv.2405.14143

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A Geometric Perspective on the Closed Convex Hull of Some Spectral Sets

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A Geometric Perspective on the Closed Convex Hull of Some Spectral Sets

Renbo Zhao

arXiv.org

Cornell University

05/23/2024

DOI: 10.48550/arxiv.2405.14143

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https://doi.org/10.48550/arxiv.2405.14143View

Preprint (Author's original)This preprint has not been evaluated by subject experts through peer review. Preprints may undergo extensive changes and/or become peer-reviewed journal articles. Open Access

Abstract

We propose a geometric approach to characterize the closed convex hull of a spectral set S under certain structural assumptions, where S which is defined as the pre-image of a set C⊆Rn under the ``spectral map'' that includes the eigenvalue and singular-value maps as special cases. Our approach is conceptually and technically simple, and yields geometric characterizations of the closed convex hull of S in a unified manner that works for all the spectral maps. From our results, we can easily recover the results in Kim et al. (2022) when the spectral map is the eigenvalue or singular-value map, and C is permutation- and/or sign-invariant. Lastly, we discuss the polynomial computability of the membership and separation oracles associated with the (lifted) closed convex hull of S.

Mathematics - Optimization and Control

Details

Title: Subtitle: A Geometric Perspective on the Closed Convex Hull of Some Spectral Sets
Creators: Renbo Zhao
Resource Type: Preprint
Publication Details: arXiv.org
DOI: 10.48550/arxiv.2405.14143
eISSN: 2331-8422
Publisher: Cornell University; Ithaca, New York
Language: English
Date posted: 05/23/2024
Academic Unit: Business Analytics
Record Identifier: 9984628158502771

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